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ACE No. LilJlO 



NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 



WARTIMK REPORT 



ORIGINALLY ISSUED 

NoTember 1944 as 
Advance Confidential Report L4J10 



THE THEORY OF PROPELLERS 
III - THE SLIPSTREAM COHTRACTIOW WITH NIJMERICAL 
VALUES FOR TWO -BLADE AOT) FOUR -BLADE PROPELLERS 

By Theodore Theodorsen 



Langley Memorial Aeronauticetl Laboratory 
Langley Field, Va. 



r- 



NACA 



WASHINGTON 

NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of 
advance research results to an authorized group requiring them for the war effort. They were pre- 
viously held under a security status but are now unclassified. Some of these reports were not tech- 
nically edited. All have been reproduced without change in order to expedite general distribution. 



L - 155 

DOCUMENTS DEPARTMENT 



Digitized by the Internet Arcliive 

in 2011 witli funding from 

University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation 



http://www.archive.org/details/theoryofpropelleOOIang 






]?ACA ACR No. Li|--T10 

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 



ADVANCE CONPIDENTIAL REPORT 



THE THEORY OF PROPELLERS 
IIT - THE 3LIPSTRE:\IV! CONTRACTION WITH NUMERICAL 
VALUES FOR TWO -BLADE AND POUR -BLADE PROPELLERS 
By Theoaore Theodorsen 



As the conditions of the ultimate wake are of concern 
Loth theoretically an^'i practically, the magnitude of the 
slipstream contraction has been calculated. It will be 
noted that the contraction in a representative case ic 
of the order of only 1 percent of the propeller diaiiieter. 
In consequence, all calculations need involve only first- 
order effects. Curves and tables are given- i'or the 
contraction coefficient of two-blade and four-blade 
propellers for various values of the advance ratio; the 
contraction coefficient is defined as the contraction in 
the diarr-eter of the wake helix In terms of the wake 
diameter at infinity. Tbe contoui" lines of the wake 
helix ere also shown at four values of the advance ratio 
in compalrison wltli the contour lines for an infinite 
number of blades. 



INTRODUCTION 

Since reference is often made to the wake infinitely 
far behind the propeller, it Is desirable to establish 
certain relationships between the dimensions of the 
propeller and those of the walce helix at infinity. The 
present paper considers the relationship of the propeller 
diameter and the vvake di am.eber, or the problem of the 
slipstream, contraction. 

Tne discussion is restricted to a consideration of 
first-order effects, that is, to the determination of the 
contraction per unit of loading for infinitely sm.all 
loadings only. It will be seen that the contractions 
are indeed very small, of th:=- order of a few percent of 



CONFIDENTIAL NACA ACR No. L^JIO 



the prooeller diameter, and that the high-order terms 
are therefore not of concern. The interference velocity 
accordingly is neglected as s^all compared with the 
stream velocity. The wake helix lies on a oerfect 
cylinder cmd the pitch angle is everywhere the saine . It 
is noted that the assujnption of zero loading corresponds 
to that used by Goldstein I'or a different purpose. 



SMBOLS 

R tip radius of oropeller 

r radius of elerrsnt of vortex sheet 

Ar contraction 



Ar^ total contraction or contraction at ~ = 



o 



T angle between starting point of soiral line and 
point P 

H pitch of SToiral 

G angular coordinate on vortex sheet 



advance ratio 






X ratio of radius of element to tip radius of vortex 
sheet (r/R) 

vpj radial velocity 

V advance velocity of propeller 

w rearward displacement velocity of helical vortex 
surface 

— w 

w = - 

p number of blades 
K mass coefficient 

CONFIDENTIAL 



NACA ACR No. lJ-JIO CONFIDENTIAL 



2kw 



r 

K(x) 



circulation at radius x 



TfcXVR 



K(x), 



CO 



circulation function Tor single rotation 

angular velocity of propeller, radians per 
second 



ZttVw 



radial velocity at Doint P due to a doublet 
element at 9,x except for a constant factor 



/ 



/ _\b cos (e+T^ - s^.n (9+tV Ti- 2x^+\2e2+ ^ cos (S + t)] 
Yl - 



\ 



1 + x2 + \292 _ 2x cos (9+t) 



5/2 



72 = 



Yi = 



^2 



_ K(:0 



where n = 0, 1, 2,..op-l 



n 



/'I 



'JO 



Jo dx 
angle of contraction, except for a constant 



Y, 



factor Y2 = / Y-|_ d9 
contour line of contraction, except for a 



constant factor Ix h- fy^ = ' Yp d9j 

K LL ' ' 



\ ' '70 



Ar 



o 



R 



/ c - > 5 ^ 

total contr'action in terms of radius ( — ^ -^ Y ' 



K k 



■5 



X3 



— contraction coefficient -r- Yz 



CONFIDENTIAL 



li- 



CONFTDENTIAL 



MAC A ACR No. L^JIO 



^ T — 



w. 






\^ 



I - k) F(k) - I E(k) 



ix=l 



J 



THEORY 



Tbe radial veloc.1ty is obtained by using the Biot- 
Savart lav and integrating ovor tno entire surface oi' 
di3ccntlnui ty. If Z^rg 



pro>:)ler;i is to determine che ratio 



is tae total CDntraction, tlie 



o 



P. 



for various 



nurii"be r s of b 1 ad e s 



several advance ratios. Simple 



expressloiiS referring to zero loading are used throughout. 



The radial 

is calculated, 
from an element 
on a spiral q:).^ 
dicular to the 
The sngle betwe 
and the noint 
belov; the plane 
is designated 
frorri the starti 
distance h be 



inv/ard 

(See fi 
of circ 
radiur 
axi s and 
en the s 
P is de 
to infi 
H, the 
ng point 
low tlie 



dVK' 



velocity 

g s . 1 and 2 . ) 



at the point P 

This velocity results 
ulatiOii f dsj which is located 
r th'it starts in a plane perpen- 

containing tlie reference ooint P. 
tarting point of the spiral line 
signated t. The spiral extends 
nity. If the pitch of this spiral 
element at a projected angle 

of the spiral is then at a 
reference plane where 



h 



T 



(1) 



c-j introducing the nondimensional quantities 



\ = - 



X = -^: 

R 



V 



(2) 



in the Biot-Savart law, the follovving expression is 
obtained for the rac'ial inward velocity dvg' due to an 
element on the vi/ake helix o± strength f r 

COInTPIDENTIAL 



KACA ACR No. iJ+JlO CONFIDENTIAL 5 

1 f d9. P cos (e + t) - sin (6 + t) ,^> 

GVj^' = Ajc — . :^ — J- (5) 

■'I'T R [l + x2 + ^20 2 _ 2^, CQS (e + t)J5A 

By di.f f erentiating equation (5) virith respect to x, 
the field of a douolet eleinent on the helical vortex 
sheet is obtained, the doublet eleinent consisting of two 
neighboring singlet elements each of strength f . 
Setting f dx equal to p and dividing tlrirough by the 
stream velocity V gives 

dlB.= 1 r d6 -^ [e PCS (9+T) - si n fG +T J [l - gx^ + \^9^ +x cos (B-t-T).! ,,. 
^' " hrr RV ■ r^ ^ ^2 , ^202 . ^x cos (e+T)]5/2 

where vj^ is the radial velocity at the point P. 
Equation ([;.) may be written in the form 

'Vr 1 r d9 

where 

_ [p cos (e+T)-sin (0 +t)1 [l - 2x'^ + X.^9^ +x cos (B+t)] ,., 

fl - -^ ^^ -^-^ 2 ^ i ° ) 

[i + x^ + \^q'^ - 2x cos (9+t)]5/^ 

The function j-i is plotted against 9 + t for four values 
of X. and various values of t and x in figures 5 to 6. 
With 

rn ^ 2rrVw ^^,^^^ 

pcO 

^ 2rr£w ^(^) 
pw 

rrVRX c 



where 

2k:w 



- K(x) (7) 



= 2kw (8) 

CONFIDENTIAL 



6 COKFIDEITTIAL NACA ACR No. li^JlO 

substitution in equation (S) gives 

alf< = . V' ^ m y, de (9) 



9 
If the T>oint P is at a distance h = H — below 

the propeller, integrating equation (9) over the wake 
yields 

IR. =.^5i I il(.xlYV, dz d9 (10) 

J-9 Jo 

It is noted that, v/ith equally spaced blades, the 
function 

X ■/! (11) 

is an odd fxinction of 9 and 

fte 



y / Vt d9 = (12) 

■:i' '.-'-9 



Equation (10) can therefore be rewritten as 

n-^ nl 



Y = -l~- j ^l.Ji^-^^ (15) 



CONFIDENTIAL 



NACA ACR No. L)+J10 CONPIDENTIAL 



7 



Let 



- n -^ 



Yt = 



1^0 



Y: 



= t 



y-T ajc 



Yi de 



i^e 



(11+) 



Values of Y-i and 



Y;., 



mrltiwlied by a constant factor 

for convenience in TDlotting, are given in tables I to IV 
for two-blade and four- blade prot'ellers for which \ 
and G take on various values. Theae functions are 
plotted against B in figures 7 and 6. 



Equation (15) becomes 






\2 c. 



Y- 



15) 



Nov; 



V 



_K 
V 



dr 
dh 



^'h 



d' 



h 



1 '1^ 
\ d9 



(16) 



Therefore 



V 



d8 



R 



\^ Os 



h 



- Yp 



(1?) 



v/hence 




Ar _ R2 - Rl . 
R R 


- 1 

R 



pR^ 



dr = 



IX 



dx = - 



\^ c. 



cfR- 



J. 



Ll k 




Yo d9 (13) 



p A Tjm T rMPT-.Tm t /i t 



8 CONPIDEIITIAL NACA ACR No. iJ^JlO 



If R2 is the radius at the propeller and R]_ is the 
ultimate radius of the wake (32 = 0, 9x = "= 1, 



Jo 



where 



Yz = / Yo de 
I/O 

Values of Yz are given in tables V and VI and are 

plotted in figure 9 f<^r t\*o- blade and four-blade propellers 
for v/hich X ar.d 9 take on various valu.es. 

After all substitutions are made, the conTjlete 
multiple integral for the total contraction is obtained 
as 



,'.'o nrc ,1 

— = - -r- I I / -v(x) ->_y. (9,x) dx d9 d9 
CO t;0 I/O '^ 

(See figs. 10 and 11. ) 

INFINITE Nm'BER OF BLADES 



For purooses of comnarison, it is useful to obtain 
the contraction for the case of an infinite number of 
blades. Ey resolving the circulation into components 
parallel to and osrpendicular to the axis of the v/ake, 
the helical vortices can be replaced by a system of 
vortices parallel to the axis and another of ring vortices 
having centers on the axis. Only the ring vortices 
contribute to the radial velocity. 

The field due to a vortex ring of strength f and 
radius r = Rx, located at a distance h belov/ the 

CONFIDENTIAL 



NACA ACR No. liiJlO CONFIDENTIAL 



reference point P, is given by Lamb (reference 1, 
p. 257)' In the notation of the present paper it is 



fR 2^/2 ,\ _,., . 2 ,.,, V 



(20) 



where ii(k) and ?(k) are the complete elliptic 
integrals and 



.2 



M-X 



h^^ 
R 



+ (1 + x)' 



As before, a doublet ring is obtained by differentiating 
equation (20) vjith respect to x. 3y setting 



f dx = 



E 



the following expression is obtained for the 'field of a 
doublet ring: 



r 1 
° Sir H I 1 - k'^ 



I ' 



2 i 

E(k) + x^ 



2 - k^ 
2?(k) - ^ ^ E(k) 

1 - k^ 



>(21) 



In order to obtain the effect of the entire vortex 
system, equation (21) is integrated, with respect to h/R 
and X as 



r.i n 



PR" 



•Jo J -h/R 



BttH 



k < X 



1 

? 




k2 




1 


- k2 



Efk) + xS 



2F(k) - 



1 - 



^^^^L 



U|dx 



(22) 



The radial velocity 



1 6v 

VR = — -^ 

R 



CONFIDENTIAL 



10 CONFIDENTIAL NaCA ACR No. IJ+JIO 

Equation { 2Z) may be v;rltten in the form 



so that 



Now 



'-'0 -J-h/R 









^0 



^r 1 / dr ,, 
— = - / ■— ah 
R t. I dh 

nh/R 

,/ V fi 



/'I ;'■•' /, N 



3y substituting 



yj (/h/R 



TTCoX-VR 

r 3: — ^- — K(.x) (26) 

where K(x) is the horizontal comnonent of the circulation 
coefficient, wiilch may be expressed as 

K(x) W--^;^— V" (27) 

CONFIDENTIAL 



MAC A ACR No. LiiJlO CONI-'IDENTIAL 



11 



the follo'wlng equation is finally obtained; 



P 



16ttK 



v\2.x2y 



x/2 1^^ 
dx I k, 



2-l_ E( k) . x2 



1 - k"^ 



2 l1 

2F(k) -^^^E(k)Hd^ 
l-k2 JJR 



(28) 



For convenience in usin,^ the Legendre tables, the 
second integral is written in the form 



■' z T d-- 
1 R 



(^9) 



(ih/R 



where 



r 



/ 1 1 



,\ 



A\ 



zi = sin ^ <j tan2|/ ^x 2 _ x2yE(k) + 2x^[p(k) 
l. 



E(k) I ^ 



_! I 



and 



/h\2 



^^^ - (1 ^ x)2 



sin^^" 



or 



0" = sin~-^k 
(See fig. 12 for plots of Z] against h/R.) 
The final ex'oression is 



Ar 



R 16ttk 



'■1/ 2 \^/2 r^ [' 

^ ^ '^ 'jh/R L 



1 A 

2 - vA) E(k) 



+ 2x2rp(k) - E(k)l !>d^ 

(See table VII and fig. IJ . ) 

CONFIDENTIAL 



12 



CONFIDEFTIAL MAC A aCR No. lJ+JIO 



INFINITE rj:.!3ER 0? BLADES FOR DUAL ROTATION 



The contraction for a dual-rotating propeller v/ith 
an infinite number of blades is next obtained. In this 
case ,K(z) = 1, and the radial velocity is 



*^R 



2ttH 



r 



,2 U^ - k) F{k) - ^ E(k) > 



x=l 



x=0 



Since the value at the lov;cr limit is zero axid K(x) = 1 
for an infinite nu"iber of blades, it follows by substituting 



Ar 

R 



^.v■ 



Jh/R 



w-i d— 
^ R 



where 



\(rj - ^) P{^) - I E(k: 



I \v. 



x=l 



(See table VIII and fig. ll;..) 



CONCLUDING RSI ARKS 



The contr 
and four-tlade 
values of the 
tri-nle Integra 
and susceptibl 
methods are de 
hoped that the 
purDose. It i 
contraction, 
and advance ra 
terms of the r 



action coefficients are given for two-blade 

single-rotating propellers at four specific 
advance ratio. The calculations involve 
tions and are therefore somewhat laborious 
e to numerical errors. Until more convenient 
vised to perform this integration, it is 

values given In this paper will serve the 
s well to notice the small magnitude of the 
A four-blade orooeller with normal loading 
tio is shown to have a total contraction in 
adius of Ifcss than oa^ percent. The 



CONFIDEICTIAL 



NACA ACR No. lIlJIO CONFIDENTIAL IJ 



first-order treatmezit embodied in the paoer is therefore 
adequate for all technical purposes. 



Langley Memorial Aeronautical Laboratory 

National Advisory Coiraiiittee for Aeronautics 
Langley Field, Va . 



RLPERSNCE 



1. Lamb, Horace; Flydrodynamics . Sixth ed., Cambridge 
Univ. Press, 1932. 



CONFIDENTIAL 



NACA ACR No. LiiJlO 



CONFIDENTIAL 



111 



TAPLE I 



FUNCTION 



\^ 



il- 



\\ FOR TVv'O- BLADE PROPELLER 



9 
(de-) 


\3 i 


p 
(deg) 




'h 




K --1 


\ = 1 


-^i 


50 


-0.0002^^ ! 


1 


-0.000133 




-0.000683 


100 


.000148 


2 


-.000213 


-0.000282 


-.001033 


150 


.001906 


I 


-.000293 




-.001367 


150 


.004815 


-.000382 

-.oocLSS 




-.001232 


170 


.000679 


5 




-.001235 


200 


.OO5OS2 




-.000R70 


-.000925 


-.001316 


220 


.002008 : 


7 


-.000^82 




-.001358 


250 


.000730 1 


p 


-.00078'^ 




-.001353 


300 


.000,'.' so i 


5 


-.000898 




-.001350 


550 


.0013U5 , 

1 


10 


-.000984 


-.001792 


-.001358 


i+OO 


.000622 


20 


-.001159 


-.001571 
-.001286 


-.001567 


500 


.000270 


ho 


-.000999 


-.001030 


550 


.oookoo 


60 


-.000802 


-.000S62- 


-.000759 


600 


.OOOli+S 


so 


-.000359 
.000^60 


-.000321+ 


-.OC0551; 


650 


.000051 


100 

1 


.000312 


.0u008'+ 


700 


.00 oil' 9 


120 


.001730 


.000709 


.000271^ 


750 


.000175 


ll.:.0 


.003151 


.OUOS09 


.00054.7 


800 


.00 00' 58 


160 
ISO 

1 200 

1 


.003351 


.00071+0 


.000521 


900 


.000079 


.002638 


.000^92 


.000152 






.ooi5ii-'-^ 


.000257 


.000100 






220 


.000625 


.000093 
.ooooili- 


.000073 






2I..O 


.000160 


-.000015 






260 


-.Q00CS6 
.ooooli.9 


.000005 


.000008 






280 


-.000013 


.000036 






300 


.000213 


-.oooool 


.0000li:7 






^20 


.000363 


.OOOOOli 


.000036 






3L0 


.000390 
.0003I1.3 


.000008 


.000050 






360 


.000051 


.000001 






380 


.000252 


.000003 


-.000003 






Uoo 


.000115 


-.000005 


.000001^ 






1,20 


.000027 


.000020 


.000007 






ldi.0 


-.000011 


6.000011 




, 


.. , . 1 


k6o 


.000011^ 







NATIONAL ADVISORY 
COT.TVJITTSE FOR AERONAUTICS 



fAfTiPTDFWTiTA''", 



NAG A ACR No. l1|J10 



GONPIDENTIAL 

5 



15 



A- 



TABLE TI.- FUNCTION -y-Yj_ FOR FOUR-BLADE PROPELLER 



n 


'i?'. 


9 
(dee) 




\5 




(deg) 


'=i 


. = i 


\ = 1 


X = il 

2 



10 
50 
60 


0.000500 

-.0206S0 

.097700 

1.251120 

1.'?5S120 


1 
2 

'5^ 


-0.001+211 


-0.000001 


-0.000275 

-.000250 

-.000088 

.000083 

.000251 


70 

80 

65 
90 


2.879450 

3. 81; 9270 

,'+.59L.6oo 
)+. 967600 
i)-. 538300 


6 

9 

10 


-.011371 


.000002 
-.000013 


.000501 
.ooofoij. 
.000905 
.001263 
.0011+32 


95 

100 
110 
120 
160 


3. 5^! 9380 
2.685010 
1.^30560 
.821510 
1.017150 


20 

1+0 

60 

80 

100 


.006150 
.I57OOI1. 
.563018 
.3I-12699 
.2185I1.8 


.001786 
.0061+66 
.006i+2^ 

.ooiii7l+ 
.002161 


.002981+ 

.00 5 5 28 

.005108 
.001577 
.000552 


200 
250 
^00 

350 

1+00 


.902770 
.i4.3696o 
.523530 
.217700 
.15kl90 


1 120 

1 lixO 
160 
180 
200 


.112755 
.091686 

.075117 
.053218 
.O4O86I 


.001281 
.000865 
.0005^5 
.0005LS 
.000221+ 


.000293 

.000221+ 
.00 02 1I+ 

.000166 
.000073 


i+50 

500 

600 


.15I4.OIO 

.119570 

.038I150 

-.029500 


220 

^)'0 

260 
280 
300 


.050025 

.0211839 
.021139 
.017093 

.0lll.0l|8; 


.000176 
.000130 

.000100 

.000062 

.0000l,'.2 


.000101 
.000066 

.oooos? 

.0000I+8 
.000037 






320 
3I4-O 
560 

1+00 


.0111714. 
.009069 
.007762 

.0061+39 

.001+650 


.000052 

.000016 

.000052 

-.000018 

-.00001:1 


.000025 

.000002 
.OOOOOlj. 






1+20 

J+ko 

[(60 
1 


.0037i|8 
.00588^. 
.oorjJ+28 


-.00000^ 

-.000001+ 





NATIONAL ADVISORY 
COMMITTEE FOR AERONAUTICS 



CONFIDENTIAL 



NACA ACR No. L/+J10 



CONFIDENTIAL 



16 



TABLE III.- .PTINCTION 



\ 



h'' 



FOR TWO -BLADE PROPELLER 



e 


%^ 


( 

e 


1 ' 


(deg) 


>-.i 


(deg) 


^ = 1 


\ = 1 


X= li 




^ 


! 


2 




2 





8.138900 


1 

i 


0.377250 


-0.00)w'i9l4 


-0.003156 


20 


8.159900 
8.219I1.00 


i ^ 


.377650 
.373580 


-.ooM+55 




1+0 


^^ 


-.0014.280 




60 


8.29llj.00 


1 6 


.^8011^0 


-.003990 




80 


8.351)4.00 


i 8 

1 


.382320 


-,005Il90 




100 


8.^58900 


\ 10 


.385170 
.C02ii2C 


-.002790 


-.002[;.66 


120 


8.223900 


20 


.000595 


-.001651 


iLo 

loO 


7'7k5300 


p 






-.000876 
-.000139 


6.570900 


1.0 


.U37620 


.006355 


18 


4.917900 


i 50 

1 






.OOOI1O7 


200 


3.iLi2900 


1 60 


.U67020 


.010675 


.000907 


220 


2.512900 


30 






.001302 


2l|0 
2o0 


2.232900 


.1.86770 


.013075- 


.001577 


2.202900 


90 






..001717 


280 


2.21.5900 


100 


.L.85670 


.013075 


,001724 


300 


2.215900 


120 


.L.53070 


.011070 


.001504 


320 


2.026900 


1.80 


.573150 


.008050 


.OGII29 


5JaO 


1.7^^900 


loO 


.267250 


,001+870 


,000718 


5oO 


I.k0li900 

i.ioSIiOO 


i3o 


.169!: 00 


.002li70 


.000[>li)| 


380 


i 200 
1 


.102200 


.001070 


.000313 


I|.00 


.890900 
.78014.00 


1 

i 220 


.067520 


.oooii70 


.000207 


1x20 


2l;0 


.055?io 


.000280 


.000178 


ao 


. 7ii5L.oo 


260 


.055i.).6o 


.000270 


.000 18 9 


1+60 


.7Lol^'oo 


280 


.05281Q 
.01,8820 


.000280 


.000162 


500 


.671.1+00 


i 500 

1 


.000290 


.000112 


5Uo 


.50114.00 


! 5f^o 


.0/4.0500 


.000290 


.000065 


530 


.32^1^00 


5lp 

560 


.02863c 


.000260 


.000025 


620 


.2L.61i00 


.016630 


.000150 


.000007 


660 


.2^lliOO 


^30 

j 


.007610 


.000090 


.000007 


700 


.18914,00 


.0021^00 


.000080 


.000006 


7'fO 


.112000 


lj.20 


.OOOI42O 


.000070 




780 


.030000 


kko 


.OOOlOu 






£20 


-.006000 










660 


-.OOliOOO 

_ 












CONFIDENTIAL 



NATIONAL ADVISORY 
COMMITTEE FOR AERONAUTICS 



NACA ACR No. Li|J10 CONFIDENTIAL 



17 



TABLE IV.- FUNCTION —Yp FOR POUR -BLADE PROPELLER 



T 

1 




n 








1 

9 
(deg) 


V'. 


1 

e 

(deg) 




\5 






1 


1 


1 




1 

! 




X = 1 


X = 1- 
2 





OJJdlklk 





0.017253 


0.007918 


0.005175 


10 


.0-dllhlk 


2 


.017259 


.007913 


.005107 


20 


.02i+li.lb 


ij 


.017261 


.007913 


.0014.728 


50 


.021^.380 


.017266 


.007918 


.00k08li 


ko 


.021^.256 


b 


.017276. 


.007913 


.003^1!^ 


5^ 


.02^969 


' 10 


.017287 


.007918 


.0023 1+3 


6o 


.025353 


1 ^^ 


.017^17 


.007908 


.001699 


70 


.021953 


ko 


.010676 


.006875 


.001235 


30 


.019555 


1 60 


.013899 


.001^520 


.000902 


90 


.oi6oIj.9 


i 80 


.009902 


.002668 


.000675 


100 


.0iJ5ii0 


1 100 


.00681+5 


.001591 


.OOOS14.1 


110 


.012166 


1^0 


.oo';;099 
.ooLi-Oio 


.OOlOlli. 


.oooI'04 


130 


.010889 


, liiO 


. 000 bl,lj. 


.00031!^ 


150 


.009821 


1 160 


.00^112 


.0001,07 


.0002L.0 


170 


.U08lfi4Jx 


iGo 
j 


.002383 


,000255 


.000173 


190 


.0069SI 


1 

200 
220 


.001859 


.000157 


.000135 


210 


.00S705 
.00^773 


.001^86 


.000091 


.00U102 


230 


21,0 
260 


.001186 


.000087 


.000075 


250 


.ooLo92 


.0009119 
.0007i.4 


.OOU01I.9 


.000052 


270 


.00352B 


2SO 


.000028 


.000035 


290 


.00303s 
.002594 


1 ^;00 


.ooo;3q6 


.000010 


.000020 


510 


5f-0 


.000i,72 


-.000002 


.OOOC09 


530 


.002210 


1 3i-o 


.000363 


-.000010 


.000005 


550 


.001337 


1 360 


.000^71 


-.000019 




570 


.001612 


1 530 


.000192 


-.000021 




390 
lio 


.001576 


koo 


.00U132 


-.000015 




.001166 


I;20 


.000087 


-.OOOl^'OI 




{^50 


.000959 


.oooo1l6 






L.50 


.00075I; 


1 








I170 


.000550 


i 
1 


• 






U90 


.000567 


1 








510 


.000205 


1 








550 

1 . 


.000075 


1 

i _-i 









1-7 AT 1 NAL AD VI SOR Y 
COMMITTEE FOR AERONAUTICS 
CONFIDENTIAL 



NAG A ACR No. L'iJIO 



CONFIDENTIAL 



18 



X 



3 



TABLE v.- l^TKCTION ~^' FOR TWO -BLADE PROPELLER 





^ ^ 
/\",. 


1 










1 -z 








9 
(deg) 


'4- -^ 


! e 

i (deg) 




k - 




i_ 


X =^ 


\ = 1 


v = il 




k 


1 
1 


2 




^ 





.o[,c6i5 


i 




o.oi99J4i+ 


0.002li^.l 


O.OOG602 


20 


.0U739 


2 


.019800 


.00215],^ 




Uo 


.cko85Q 


k 


.015656 


.002163 




6o 


.0^6926 





.01S511 


.002180 




So 


,03 29';^ 5 


8 


.015566 


.002191 




100 


,029007 


10 


.01^,221 


.002201 


.000714.3 


120 


, 0250^)6 


20 


.oi!^)'7li. 


.002217 


.000855 


lM.0 


.0212I18 


?^ 






.000921 


i6o 


.017300 


ho 


.01291$ 


.002111 


.0009SO 
.00094J.t 


l3o 


.0] 50^.7 


50 






200 


.013062 


60 


.0x1197 


.001852 


.000910 


220 


.011670 


80 






.000854 


21^0 • 


.0105U9 


.0093S2 


.OOII4.9O 


..000780 


260 


.009301 


90 






.000695 


2'^'0 


.,oo8al:£ 


100 


.0075^5 


.00109:^ 


.000607 


300 


.0073^2 


120 


.00^752 

.00l,.l'^-8 


.000728 


.0001^39 


$20 


.00 6$ 81 


140 


.0001+37 


.000303 


3iL0 


.C05l;&3 
.C0ii7~^If 1 


160 


.002927 


.0002^3 


,000208 


560 


180 


.002099 


.000136 


,oooil+8 


5 So 


.ooi;i39 1 


200 


.00159k 


.000087 


.OoOlll 


l+on 


.00366L 


220 


.001277 


.00006s 


.OOC086 


Ii20 


.003208 


2I4O 


.OOIOL9 


.000055 


.000066 


llliO 


.002003 

.002.1^9 


260 


.000837 


.0000117 


.O00Ql|8 


UbO 


280 


.000632 


.OOG039 


.000030 


300 


.001.8^2 


500 


.OOOli-il 


.000031 


.000017 


5I1O 


.001^18 


^20 


.000276 


.000022 


.000007 


5.:0 


.000 9 22 


1 -^ko 


.00Gl.'x6 
.0000 06 


.00c 018 


.000002 


620 


.cooS'-^i 


360 


,000003 


. CO 00 i 


660 


.000^.25 


^80 
Hoo 


.000021 


.OOC005 


.000001 


700 


,000232 


.000005 


.000002 




7:u0 


1 

.00008s 


Uo 


.000001 






730 


-.oooocS 










320 


-.C000U5 











NATIONAL ADVISORY 
C0:.1MITTSE FOR AERONAUTICS 



NAG A ACR No. liiJlO 



CONFIDENTIAL 
3 



19 



TABLE VI.- FUNCTION '•■— Yz FOR F01TR-3LADE PROPELLER 





'^^ i 










p 
(deg) 


.- .1 

, 1 ' 


(deg) 




L^5 








1 




\ = ± 1 

^ 1 




\ - - 

2 


X = 1 


X = 1- 





0.0677&6 





0,0^^5'^2 


0.010515 


0.005286 


10 


.06332s 

.059264 
.05S006 


2 


.0"';2969 


.010256 




20 


4 


.052567 


.009900 




?o 





.05176]! 


.009684 




.050757 


,q 

-> 


.051162 


.009407 




30 


• 0li:sl9 i 


10 


.0"'^0553 


.009151 


.ooi+539 


?0 


.0i|2l;20 1 


20 


.027537 


.007750 


.005552 


V^ 


.0331,50 i 








.U02760 


30 


.03)^2^2 1 


.02i531i 


,005153 


.002151 




.031732 


50 






.001650 


100 


.029167 


60 


.016211 


.005155 


.001297 


110 


,026892 


70 






.OOlofiO 


150 


.022S85 


80 


,012055 


.001916 


.000856 


150 


.019265 1 


90 






• .000722 


170 


.016059 


100 


.009157 


,001185 


.000618 


190 


.013358 1 

.oiiili.8 1 


120 


.007100 


.000758 


.ooo.'i65 


210 


I'lO 


.005512 


.0001^5^ 


.000510 


230 


,009505 1 


160 


.UCL,26S 

.0C351[l 


.000267 


,0002li.5 


230 


.0077^-5 ! 


i3o 


.C00151 


.000175 


270 


.0061155 


200 


.00257/-:- 


.000086 


.000122 


290 


.003288 i 


220 


.002005 


,000052 


.000080 


310 


.004^00 1 


2iL0 


.0015L5 


.000022 


.000014.9 


330 


.C03I.61 


260 


.001165 


-.000000 


.U00029 


530 


.0027l;6 


230 


.000865 
.00065!!. 


-,000012 


.U00015 


570 


,00215s 


500 


-.000019 


.000006 


390 


,001621 


^20 


.oooi.52 


-.000021 


.000002 


.ooiiai 1 


lh-0 


.000507 


-.000019 
-.COOOlf; 




[00 


,000325 1 


560 


.000196 




^50 


,0005^2 


p3o 
LOO 


.OC0116 


-,000003 




kJO 


.000306 


.000061 


-,000002 




k90 


.000155 


L^O 


.000023 






510 


.000052 i 











N.J^lTIONAL ADVISORY 
C0I.TITT2E FOR AERONAUTICS 
CONFIDENTIAL 



MCA ACR Ho. rJ+JlO 



CONFIDENTIAL 



\-^ 

K| 
Oh 
O 
PI 
Ph 

C5 

ta 
M 
EH 

<! 

EH 

O 
I 

n 



M 



o 

Eh 



O 

o 



M 

M 



en 

Eh 





r-H| ^] 


\ 

(M "3 --.C CM L~- 


^0 o-co o 




rH " 




M\-ri r.j u^r,0 


f\J O vO rCN 




"^ o 




O rH AJ N-A_:t 


^-rH ^-^f^ 






11 


O O O O O 


O rH rH 'M 




^ p^ 


*^.* 


o o o o o 
o 


O O O (3 

« ■ • ■ 








u> 




.0^ 


O O -O '■~J CO 


-d-ovo-j- 




\ 








1-M 


_-J-MA'Aj CM rH 


r-^ ,H 




, 




erico -Al_-j'-vi) 


i>--0 '^M rj KA 


r-OjO^ 


rH " 


rH 


lTnCO CT.OvJ ^ 


rrv^-rH rH \r\ 


ro>tX) 


'^ O 




O rH Al _\KO 


rJ On 




11 


o o o o o 


O rH M OJ K-v 


-t-^ 


^« 




'3 o o o o 


o o o o o 


O O 


^ 






B • 






o 












iM 


r 

h/R 


O OMD r\)a) 


-J-OOTVO-J- 


-A) rH 


-ri K^AJ rg rH 


r i rH 


, 






C M3 O OACO 


vT/ t^ONt^rvJ 


tr-HTo 


MJ " 


rHi'AJ 


-o oj 1 rv---- n:\ 


VOVO a-wt rH 


a\o.) 


"; O 




O '^J tOiU^iCO 


(AJ fj\-:i(\| rr\ 


cJ^^r^ 




II 


o o o o o 


r-l r-\ rvj KV^- 


uAr-- 


^. - 




o o o o o 


o o o o o 


o o 


^ 


O O O B • 


o e » • e 


« • 






o 














^^ 


o o~o f\ico 


—J^ O CO ^O _z)- 


OJ rH 


\. 










^ 


--rJ-KNOJ Al iH 


rH rH 




1 
1 




Lr\K-N[^a:)vo 


ra MTN 




■-•lO 


rH|-d- 


_:j vO _d-_Lj' a\ 


a. K^ 






o r\i r»-r-i o- 


o-^ o- 






1.1 


z-t a <':! r-i, , \ 


ai_:J- 




^K 


f< 


Cjr:> o <Zj Ci 
o 


o o 








LfA 




<■<- 


o O O vo f^l 


a; _.:!■ 




~k 




• • 




• 


-tJ-NACM rH rH 




1 
1 



20 






M 




O 




M 




EH 




l:^ 




< 




» 


>H 


CJ 


K 


Hi 


o 


LxJ 


oO 


«rij 


M 




;>• 


crt 


O o 


<fc 


i-q 


w 


<s w 


'rs; Eh 


O Eh 


M 


M 


H 


S 


< 


*=r 


'^ 


b 




o 



C01i?IDEKTlAL 



NACA ACR No 



L4J10 C0N7IDENTTAL 
TABLE VIII.- CONTOUR LINES - 



21 



DUAL-ROTATING PROPELLER 





h/R 


Ar 1 

R Cg 




10,00 


0.000060 






0.50 


.OOOll+l 






9.00 


.0002lj.'| 






8.50 


.000563 






6.00 


.000513 






7.50 


.0006S0 






7.00 


.000367 






6.50 


.001071L ■ 






6.00 


,00lk72 






5.50 


.001898 






5.00 

14.. 50 


.002355 






■ .002873 






if. 00 


.00^.716 
.001^671 


• 




5.50 






5.00 


.006183 






2.50 


.008839 






2.00 


.0i';390 






1.75 


.016370 






1.50 


,020350 






1.25 


.025320 






1.00 


. 05^1^90 
.oliIf9?o 






.75 






.50- 


.Oo25i+0 






.-^h5 


.066120 






.ko 


.0703li0 






.55 


.075070 






.$0 


.086I4OO 






■ 2.3 


.086570 






.20 


.0^5230 






.15 


.101230 






.10 


.110630 






.05 


.125220 





'NATIONAL ADVISORY 
COMMITTEE FOR AERONAUTICS 
COtlPIDENTIAL 



NACA ACR Ho. L4J10 



CONFIDENTIAL 



Fig. 1 




NATIONAL ADVISORY 
COMMITTEE FOR AERONAUTICS 



CONFIDENTIAL 



Figure 1.- Geometric relationships of wake helix* 



NACA ACR No. L4J10 



CONFIDENTIAL 



Fig. 2 



rde 




NATIONAL ADVISORY 
COMMinEE FOR AERONAUTICS 



CONFIDENTIAL 
Plgvire 2.- Plan view of wake helix showing geometric relationships, 



NACA ACR No. L4J10 



Fig. 3a 



jij 
































'" 
















-. 


■~^ 






-f-fi 






~ 




~ 




'"■ 


— 


^'\. 








































































































































v^^ 


V, 










































































s 


t^^ 










^ 






1 




























































N 


'0, 








g 






















' 














































\^ 








a 








• 




























































\^ 


\ 




5l 


^ 




































































1 


1 




i 


a 






y 


I 








































: i 




















.' 


/ 




s 








V, 


r. 




■ 






























: ;■ 


























'/' 








i 








: ' 


























































A 


V 














!; 


























































',; 


w 
















■;•■-■ 


























.' 












V 


















, 


/ 






































































'■' 


,'' 


/ 




















'h- 






































: 1 












/ 


/ 


y 
















' 












































- 










/ 




/ 
























:■:•■ 


' 














































\ 




k 1 
























% 


1 














































1 


\ 


^ 


il 






















i 




^. 






.^ 






■' 






'i 












'■ ; 


















s 


^^ 












1 1 
































































V 


\ 




















< 

y — 1 

W 
Q 

O 






























- 


' 






















"1 


^ 






































































«^ 




































••^; 




t.'l 














' i- 




















^ 










































































N\ 


\ 






























;- 








































^' 


\ 




























11 










































1 


IJ 






























,! 








































/ 


* 










1 














. 










, 


































/ 


/ 


/ 


































































/■ 


''/ 


' f 


































































f 1 


1^ 














h' 






' 
























' 






















-r 


."" 


&'' 


j^ 


















ill- 
















































•J 


^ 


























ft^v 


I": - 
























„ , 


_ 


-- 


-• 


^ - 


'^■ 










-• 


' 






,"'. 






















k . 




^^ 




-■ 


>— 


-- 


-- 




"" 


' 




^ 




T" 








^ 


>- 


^ 


































>*_ 






















f 














r 






. 


































'V 


-. 




















'^ 


^^ 












V- 








































'n' 










-" ; 


— 


— 


h- 




r- - 










-- 


' — 








*^ 


^ . 


































"-^ 


































■*■ 




- . 




L- , 


. ^ 


• •» 


•*- 


-, 


























!*'• 


















■-- 




























■"• 


-J 




•"t. 


^ 


- ^ 




















i'^ 


■ . 




















































" ■» 


S. 


"* ■* 


<. 


. 






































































*■ 


\ 




- 


•N 












p"' 
































































\ 


\ 










% 






































-!,- 




















] 






\ 




\ 














































;l.; 




















?t 






1 




\ 






— 


tt'i 










-" 
























''■ : 
























d 






\ 










































































\ 


1 










S:" 


'-'-i 
































', ' 


1-^ 




, 1 


























N 


^J 








s 














































'til 
























r 






'[j; 


il. 
















1'.! 


















































/ 


1 


1 




i 










































> 
















?, 






^ 


X 










V' 


























































II 




f" 








































































V 



















O 

1 — I 

o 



Vl 



■3 

> 



•H 



NACA ACR No. L4J10 



Fig. 3b 



-"i ■ |- ' ■' ■ '^^1 ■ =^:fi' j ,■ ' 1 jJu";-*^!':' :'-!"" •": 


■ 


i 


_^ ^ 


\ 


8 T 


■ - 1 ' ' ■ 'E i ' : ■ ; ^^ 


Il3 ' I 


■ " _;1 j "" 


±' ■ ' - ' '- '■ li '-7 




1 ' 


, -*, ^ 2^ 


i- ■ " -^ 


^^ 


. . .: iT^ 


uJX- . 


^fc 


t!, ■ ^75.' ■ , ^ 


T ' ^ "5;; 


„_ . p: . - - ■ ^^ 


.< ■ + ^ 


• ^ ^ ^ 


. ^ it- ■ - Jl -r- i- 


M :.. ^. -:;■■. t'. 


a ±:il:"^ ' ~" "' ^ . i: 


.;: - ^ t 


. s ■ . ^ ' ■ ' t • 


o -it r 


5^ . ._ .. . ^^ " 


y ' 


J^ ' " -^"^ 




"" _.-,- — 


iT'—i:: : z T 




■■~ -11.':^— -^ -v •' 


""^=^'-- 


- ■ *!;> - 


_ ^ 


U- -^ X 


ty 




«j 1 




1 


■ . V 


<,\ 


1^ 


% ' 


II / 





_1 
< 



td 
i-J 

l-H 

O 



NACA ACR No. L4J10 



Fig. 3c 





















... 










































































\ 


















































t 


















































1 










































§ 
















































g 


g 
















































P 


















































*" 


g 
















































g 


H 
















































^ 


E 












































































































/ 


















































/ 
















































/ 


/ 
















































* 


















































\ 


















































\ 




















































^>v 


















































\ 


















































< 


















































1 








< 

1 — 1 
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UNIVERSITY OF FLORIDA 



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